Finite Group Databasebeta

← #206913

Subgroups — #206913

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#9881C214yesmaximal, complemented
5#9880C214yesmaximal, complemented
6#7915C212yesmaximal
7#7915C212yesmaximal
8C22×#125C411yescharacteristic
9C23×C42C2211yescharacteristic
10C22×C4⋊C8—21no—
11C2×#407C2212yes—
12C22×C22⋊C8C2212yes—
13C2×C4×M16—21no—
14#2851—21no—
15#2851—21no—
16#2851—21no—
17#2851—21no—
18#2851—21no—
19#2851—21no—
20#2851—21no—
21#2851—21no—
22C2×#389C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#387C2212yes—
25C2×#387C2212yes—
26C2×#385C412yes—
27C2×#385C412yes—
28C2×#382C411yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1728—21no—
40C2×#336—21no—
41#1280C2218yes—
42#1280C2218yes—
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48C2×#125—21no—
49C2×#125—21no—
50C2×#125—21no—
51C2×#125—21no—
52C2×#125—44no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C42—44no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—44no—
79#407—24no—
80#407Q818yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—28no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86C4×M16—44no—
87C2×#97—22no—
88C2×C4×C8—22no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#382—21no—
114#382—21no—
115#382—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135C25C22×C411yescharacteristic
136C23×C4—24no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—44no—
170C2×M16—44no—
171C22×C8—24no—
172C22×C8—24no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—44no—
176#125—44no—
177#125—44no—
178#125—44no—
179#125—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—28no—
241C22⋊C8—24no—
242C22⋊C8—28no—
243C22⋊C8—44no—
244C22⋊C8C2×Q818yes—
245#97—44no—
246C4×C8—44no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24C2×C22⋊C411yescharacteristic
266C24C2×C22⋊C411yescharacteristic
267C24C4×Q814yes—
268C24C23×C411yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—416no—
327C22×C4—416no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C4×Q814yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4#12412yes—
350C22×C4#12412yes—
351C22×C4C4⋊Q812yes—
352C22×C4C4⋊Q812yes—
353C22×C4#13114yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4112yes—
358M16—416no—
359C2×C8—28no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—88no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42#9914yes—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—28no—
417C23—28no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23#39112yes—
424C23#39112yes—
425C23#41312yes—
426C23#41312yes—
427C23C22×C22⋊C411yescharacteristic
428C23C2×C4×Q812yes—
429C23C2×#13112yes—
430C23#56812yes—
431C2×C4—28no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4#39112yes—
495C2×C4#39112yes—
496C2×C4#40812yes—
497C2×C4#40812yes—
498C2×C4C2×#9912yes—
499C2×C4C2×#9912yes—
500C2×C4C2×#9912yes—
501C2×C4#41012yes—
502C2×C4C2×C22⋊Q812yes—
503C2×C4C2×#12411yescharacteristic
504C2×C4C2×C4⋊Q811yescharacteristic
505C8—48no—
506C8—84no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—416no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22#199814yes—
528C22C2×#39111yescharacteristic
529C22C2×#40811yescharacteristic
530C22#287911yescharacteristic
531C22C22×#9911yescharacteristic
532C22C2×#41011yescharacteristic
533C22C2×#41311yescharacteristic
534C22C2×#41311yescharacteristic
535C22#296214yes—
536C4—22no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4#199614yes—
553C2—24no—
554C2—24no—
555C2—42no—
556C2#1738011yescharacteristic
557C2#1738111yescharacteristic
558C2#1739911yescharacteristic
559C2#1740011yescharacteristic
560C2#2130211yescharacteristic
561C2C2×#199611yescharacteristic
562C2C2×#199811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9881C2semidirect
5. #9880C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#28512
C2×#17972
C4×#4072
#98812
#98802
#79152